The aim of seismic data processing is to provide a reliable measure of the properties of the subsurface. Recorded seismic data is transformed into interpretable measures through a seismic processing sequence. This seismic processing sequence includes numerous steps, and these steps are categorized as pre-imaging, imaging, and post imaging processes. The imaging step transforms the recorded seismic data into an image with interpretable qualities. Spectral analysis plays an important role within the pre-imaging and post-imaging steps and is the basis of many applied processes. All filtering and convolution-based techniques, which can be as simple as a low-cut filter or elaborate as an elastic inversion, are founded on spectral theory, and correct analysis is important for success.
The pre-imaging processes include noise attenuation and regularization, among others. These pre-imaging processes assume that a one-dimensional (1D) convolutional model is applicable. Therefore, the data are the reflectivity of the subsurface convolved with a 1D wavelet, in time, where the data space is defined by recording time, and the x, and y directions. This is a reasonable approximation and is often valid within the limits of the experiment.
As pre-imaging seismic data are defined in terms of surface and time coordinates, where spectral analysis is based on the latter, no special attention is needed when applying filtering processes as temporal one-dimensional spectral analysis is truly representative of the recorded energy. The spatial wave-number components are intrinsically coupled to the temporal frequency by a dispersion relation, that is, they follow the wave equation at the recording surface.
As the 1D convolutional model simplifies many processes, this assumption is often also used for post migration processes, which is a reasonable approximation for flat geological settings. However, this approximation is very far from reality for steep or complex geological settings. Post-imaging data yield an image defined in x, y, and z. The three dimensions defining the space have no special significance with respect to one another, except for the sampling. That is, post imaging processes should be to some extent spatially isotropic.
Therefore, post-imaging seismic data is defined in terms of three-dimensional (3D) space coordinates. The seismic image represents the spatially variable reflectivity of the medium where the migration process effectively rotates the seismic wavelet to be normal to the direction of the medium reflectors. While this is the general case, it is often disregarded, and one-dimensional spectral analysis of the depth coordinate is still commonly used. This is usually performed after a depth-to-time conversion.
Assuming a convolutional model with a 3D spatially isotropic wavelet significantly increases the range of applicability of the post migration processes, applying filtering processes using this principle produces results valid for both steep and flat geological settings. However, using the assumption of a convolutional model with a 3D spatially isotropic wavelet introduces a complexity in seismic data processing, because the apparent spectral character of imaged seismic data is dependent on the velocity model, which can be spatially variable. Therefore, systems and methods for processing seismic data are desired that assume a convolutional model with a 3D spatially isotropic wavelet that compensate for spatial variability in the velocity model.